It is well-known that \(\mathsf {Tr}_{U}(-)\)  defines a functor
from \(\mathsf {Mod}\hspace{0.72229pt}R\)  into \(\mathsf {Mod}\hspace{0.72229pt}R\) . For any \(R\) -homomorphism \(\zeta : L\rightarrow X\) , \(\mathsf {Tr}_{U}(\zeta ):\mathsf {Tr}_{U}(L)\rightarrow \mathsf {Tr}_{U}(X) \)  is defined by \(\mathsf {Tr}_{U}(\zeta )(l)=\zeta (l)=\sum _{i=1}^{k}\zeta f_{i}(u_{i})\)  for any \(l=\sum _{i=1}^{k}f_{i}(u_{i})\in \mathsf {Tr}_{U}(L)\) . \(L\)  is (co)generated by \(\mathcal {U}\)  if and only if \(\mathsf {Tr}_{U}(L)=L\)  (\(\mathsf {Rej}_{\mathcal {U}}(L)=0\) ). In particular, \(\mathsf {Tr}_{\mathcal {U}}(\mathsf {Tr}_{\mathcal {U}}(L))=\mathsf {Tr}_{\mathcal {U}}(L)\)  and \(\mathsf {Rej}_{\mathcal {U}}(L/\mathsf {Rej}_{\mathcal {U}}(L))=0\) . From [1]}, if \(I\)  is a right ideal of \(R\)  and \(M\)  is an \(R\) -module, then \(\mathsf {Tr}_{R/I}(M)=\mathsf {Rej}_{I}(M)\) . For more consequences about trace and reject, we refer to [1]} and [3]}.
